CompCert is a certified C compiler which comes with a proof of semantics
preservation. What this means is the following: the semantics of the C code
you write is preserved by CompCert compilation passes up to the generated
machine code.
I had been interested in CompCert for quite some times, and ultimately
challenged myself to study Clight and its semantics. This write-up is the
result of this challenge, written as I was progressing.

CompCert has been added to
Installing CompCert is as easy as
More precisely, this article uses
Once
Our goal for this first write-up is to prove that the C function
returns the expected result, that is

## Revisions

This revisions table has been automatically generated
from the
`git`

history of this website repository, and the
change descriptions may not always be as useful as they should.

You can consult the source of this file in its current version here.

2021-03-28 | 2021 Spring redesign | 495f9db |

2020-12-08 | Advertise the version of compcert used to build this article | a611c4e |

2020-12-08 | Update to CompCert 3.8 | 26b5fbf |

2020-07-14 | Prepare the introduction of a RSS feed | c62a61d |

2020-07-12 | Add missing revisions tables | ca7a8d6 |

2020-03-20 | Add a new post about Clight and its semantics | 7073116 |

## Installing CompCert §

`opam`

, and as a consequence can be very easily
used as a library for other Coq developments. A typical use case is for a
project to produce Clight (the high-level AST of CompCert), and to benefit
from CompCert proofs after that.
opam install coq-compcert

`coq-compcert.3.8`

.
`opam`

terminates, the `compcert`

namespace becomes available. In
addition, several binaries are now available if you have correctly set your
`PATH`

environment variable. For instance, `clightgen`

takes a C file as
an argument, and generates a Coq file which contains the Clight generated by
CompCert.
## Problem Statement §

int add (int x, int y) { return x + y; }

`x + y`

. The `clightgen`

tool
generates (among other things) the following AST (note: I have modified it
in order to improve its readability).
From compcert Require Import Clight Ctypes Clightdefs AST

Coqlib Cop.

Definition _x : ident := 1%positive.

Definition _y : ident := 2%positive.

Definition f_add : function :=

{| fn_return := tint

; fn_callconv := cc_default

; fn_params := [(_x, tint); (_y, tint)]

; fn_vars := []

; fn_temps := []

; fn_body := Sreturn

(Some (Ebinop Oadd

(Etempvar _x tint)

(Etempvar _y tint)

tint))

|}.

The fields of the function type are pretty self-explanatory (as it is
often the case in CompCert’s ASTs as far as I can tell for now).
Identifiers in Clight are (positive) indices. The fn_body field is of
type statement, with the particular constructor Sreturn whose argument
is of type option expr, and statement and expr look like the two main
types to study. The predicates step1 and step2 allow for reasoning
about the execution of a function, step by step (hence the name). It
appears that
We import several CompCert modules to manipulate

`clightgen`

generates Clight terms using the function call
convention encoded by step2. To reason about a complete execution, it
appears that we can use star (from the Smallstep module) which is
basically a trace of step. These semantics are defined as predicates (that
is, they live in Prop). They allow for reasoning about
state-transformation, where a state is either
- A function call, with a given list of arguments and a continuation
- A function return, with a result and a continuation
- A statement execution within a function

*values*(in our case, bounded integers).
Putting everything together, the lemma we want to prove about f_add is
the following.

Lemma f_add_spec (env : genv)

(t : Events.trace)

(m m' : Memory.Mem.mem)

(v : val) (x y z : int)

(trace : Smallstep.star step2 env

(Callstate (Ctypes.Internal f_add)

[Vint x; Vint y]

Kstop

m)

t

(Returnstate (Vint z) Kstop m'))

: z = add x y.

## Proof Walkthrough §

Ltac smart_inv H :=

inversion H; subst; cbn in *; clear H.

We can now try to prove our lemma.

Proof.

We first destruct trace, and we rename the generated hypothesis in order
to improve the readability of these notes.

smart_inv trace.

rename H into Hstep.

rename H0 into Hstar.

This generates two hypotheses.
In other words, to “go” from a Callstate of f_add to a Returnstate,
there is a first step from a Callstate to a state s2, then a succession
of steps to go from s2 to a Returnstate.
We consider the single step, in order to determine the actual value of
s2 (among other things). To do that, we use smart_inv on Hstep, and
again perform some renaming.

Hstep : step1 env (Callstate (Ctypes.Internal f_add) [Vint x; Vint y] Kstop m) t1 s2 Hstar : Smallstep.star step2 env s2 t2 (Returnstate (Vint z) Kstop m')

smart_inv Hstep.

rename le into tmp_env.

rename e into c_env.

rename H5 into f_entry.

This produces two effects. First, a new hypothesis is added to the context.
Then, the Hstar hypothesis has been updated, because we now have a more
precise value of s2. More precisely, s2 has become
Using the same approach as before, we can uncover the next step.

f_entry : function_entry1 env f_add [Vint x; Vint y] m c_env tmp_env m1

State f_add (Sreturn (Some (Ebinop Oadd (Etempvar _x tint) (Etempvar _y tint) tint))) Kstop c_env tmp_env m1

smart_inv Hstar.

rename H into Hstep.

rename H0 into Hstar.

The resulting hypotheses are
An inversion of Hstep can be used to learn more about its resulting
state… So let’s do just that.

Hstep : step2 env (State f_add (Sreturn (Some (Ebinop Oadd (Etempvar _x tint) (Etempvar _y tint) tint))) Kstop c_env tmp_env m1) t1 s2 Hstar : Smallstep.star step2 env s2 t0 (Returnstate (Vint z) Kstop m')

smart_inv Hstep.

rename H7 into ev.

rename v0 into res.

rename H8 into res_equ.

rename H9 into mem_equ.

The generated hypotheses have become
Our understanding of these hypotheses is the following
The Hstar hypothesis is now interesting
It is clear that we are at the end of the execution of f_add (even if Coq
generates two subgoals, the second one is not relevant and easy to
discard).

res : val ev : eval_expr env c_env tmp_env m1 (Ebinop Oadd (Etempvar _x tint) (Etempvar _y tint) tint) res res_equ : sem_cast res tint tint m1 = Some v' mem_equ : Memory.Mem.free_list m1 (blocks_of_env env c_env) = Some m'0

- The expression _x + _y is evaluated using the c_env environment (and we know thanks to binding the value of _x and _y), and its result is stored in res
- res is cast into a tint value, and acts as the result of f_add

Hstar : Smallstep.star step2 env (Returnstate v' Kstop m'0) t0 (Returnstate (Vint z) Kstop m')

smart_inv Hstar; [| smart_inv H ].

We are making good progress here, and we can focus our attention on the ev
hypothesis, which concerns the evaluation of the _x + _y expression. We
can simplify it a bit further.

smart_inv ev; [| smart_inv H].

rename H4 into fetch_x.

rename H5 into fetch_y.

rename H6 into add_op.

In a short-term, the hypotheses fetch_x and fetch_y are the most
important.
The current challenge we face is to prove that we know their value. At this
point, we can have a look at f_entry. This is starting to look familiar:
smart_inv, then renaming, etc.

fetch_x : eval_expr env c_env tmp_env m1 (Etempvar _x tint) v1 fetch_y : eval_expr env c_env tmp_env m1 (Etempvar _y tint) v2

smart_inv f_entry.

clear H.

clear H0.

clear H1.

smart_inv H3; subst.

rename H2 into allocs.

We are almost done. Let’s simplify as much as possible fetch_x and
fetch_y. Each time, the smart_inv tactic generates two suboals, but only
the first one is relevant. The second one is not, and can be discarded.

smart_inv fetch_x; [| inversion H].

smart_inv H2.

smart_inv fetch_y; [| inversion H].

smart_inv H2.

We now know the values of the operands of add. The two relevant hypotheses
that we need to consider next are add_op and res_equ. They are easy to
read.
We can simplify add_op and res_equ, and this allows us to
conclude.

add_op : sem_binarith (fun (_ : signedness) (n1 n2 : Integers.int) => Some (Vint (add n1 n2))) (fun (_ : signedness) (n1 n2 : int64) => Some (Vlong (Int64.add n1 n2))) (fun n1 n2 : Floats.float => Some (Vfloat (Floats.Float.add n1 n2))) (fun n1 n2 : Floats.float32 => Some (Vsingle (Floats.Float32.add n1 n2))) v1 tint v2 tint m1 = Some res

- add_op is the addition of Vint x and Vint y, and its result is res.

res_equ : sem_cast res tint tint m1 = Some (Vint z)

- res_equ is the equation which says that the result of f_add is res, after it has been cast as a tint value.

smart_inv add_op.

smart_inv res_equ.

reflexivity.

Qed.